The spectra of the unilateral shift and its adjoint

In this note I am writing out some of the material from Paul Halmos, Hilbert Space Problem Book, on shift operators. The reason I’m doing this is because shift operators are standard objects in operator theory and every analyst should know their properties and their spectra. A reference to Problem n of Halmos is a reference to Problem n in this book. An orthonormal basis for a Hilbert space is an orthonormal set whose span is a dense subset of H. The dimension of a Hilbert space is the cardinality of an orthonormal basis for H. It is a fact that a Hilbert space is separable if and only if its dimension is countable (Problem 11 of Halmos). Let N be the set of nonnegative integers.